The generalized Tate diagrams of the equivariant slice filtration

Speaker: Guoqi Yan – University of Notre Dame

Abstract: The equivariant slice spectral sequence is invented by Dugger and is generalized and utilized to a great extent by Hill–Hopkins–Ravenel in their resolution of the Kervaire invariant one problem. Later, because of the work of Hahn–Shi and Meier–Hill–Shi–Zeng, the equivariant slice spectral sequence can be used to compute the homotopy fixed point of Lubin–Tate theories. In this talk, I will talk about equivariant computations around the slice spectral sequence using the generalized Tate diagrams invented by Greenlees–May. I will first talk about my own work on computations of the $E_2$-pages of the slice spectral sequence and its localized variants. Then I will talk about my joint work with Yutao Liu and XiaoLin Danny Shi on exploring the effect of applying the generalized Tate diagrams on the slice filtration. We extend the transchromatic phenomenon and vanishing lines proved by Meier–Shi–Zeng for connective HHR theories to the periodic versions. Using the Tate spectral sequence, we illustrate the duality of differentials in the slice spectral sequence. This enables us to go back and forth between differentials in the positive and negative cones.